Gröbner bases and generalized sylvester matrices

Abstract

In his PhD thesis [1], Buchberger introduced the notion of Gröbner bases and gave the first algorithm for computing them. Since then, extensive research has been done in order to reduce the complexity of the computation. But nevertheless, even for small examples the computation sometimes does not terminate in reasonable time.There are basically two approaches for computing a Gröbner basis. The first is the one pursued by the Buchberger algorithm: We start from the initial set F , execute certain reduction steps (consisting of multiplication of polynomials by terms -called shifts -and subtraction of polynomials) and due to Buchberger's theorem, which says that the computation is finished if all the s-polynomials reduce to zero, we know that after finitely many iterations of this procedure we obtain a Gröbner basis of the ideal generated by F . The second approach is to start from F , execute certain shifts of the initial polynomials in F , arrange them as rows in a matrix, triangularize this matrix and from the resulting matrix extract a Gröbner basis.In project DK1 of the Doctoral Program, which was proposed by Buchberger, we pursue the second approach and seek to improve the theory in order to speed up the Gröbner bases computation. This approach has been studied a couple of times in the past, but never thoroughly. The immediate question is: Does there exist a finite set of shifts such that a triangularization of the matrix built by these shifts yields a Gröbner basis and, if so, how can we construct these shifts? We give first results in answering this question. In the following, let K be a field.In the univariate case, Gröbner bases computation specializes to gcd computation. In [3] (see also [4] for a good overview on this topic), Habicht establishes a connection between the computation of polynomial remainder sequences and linear algebra. More specifically, the problem of finding a gcd of two polynomials f, g ∈ K[x] with degrees m and n, respectively, where m ≥ n, can be solved by triangularizing the matrix

Gröbner Bases of Modules and Faugère’s $$F_4$$F4 Algorithm in Isabelle/HOL

“…Indeed, browsing through the ample literature on the subject one quickly realizes that a lot more properties of Gröbner bases, ways of computing them, generalizations, and intriguing applications could be added to the corpus of formal mathematics in the future; examples include Gröbner bases over coefficient rings that are no fields, elimination orders and their applications, converting between different term orders, and non-commutative Gröbner bases. We plan to contribute to this endeavor by formalizing the very recent approach of computing Gröbner bases by transforming Macaulay matrices (or generalized Sylvester matrices) into reduced row echelon form [35] in Isabelle/HOL. This approach has similarities to the F 4 algorithm but only computes the reduced row echelon form of one big matrix, instead of doing this repeatedly in every iteration of a critical-pair/completion algorithm.…”

Mathematical Theory Exploration in Theorema: Reduction Rings

“…There are many possibilities for future work. On the theory level, other aspects of, and approaches to, Gröbner bases (again in the original setting) could be formalized, for instance the computation of Gröbner bases by matrix triangularizations [17]. For this, the further improvement of the tools described in Sect.…”

Optimizing Elimination Templates by Greedy Parameter Search

“…However, in computer vision, polynomial systems often have the same support for different values of their coefficients. Then, it is efficient [11,58] to construct T a by (i) building a Macaulay matrix M using a fixed procedure -a template -designed in the offline phase, and then (ii) produce T a in the online phase by the G-J elimination of M [10,29]. Our main contribution, Sec.…”